Vol. 301, No. 1, 2019

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Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Leandro Antunes, Valentin Ferenczi, Sophie Grivaux and Christian Rosendal

Vol. 301 (2019), No. 1, 31–54
DOI: 10.2140/pjm.2019.301.31
Abstract

Megrelishvili (2001) defines light groups of isomorphisms of a Banach space as the groups on which the weak and strong operator topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the point of continuity property (PCP) is light. We investigate this concept for isomorphism groups G of classical Banach spaces X without the PCP, specially isometry groups, and relate it to the existence of G-invariant LUR or strictly convex renormings of X.

Keywords
groups of isomorphisms of Banach spaces, isometry groups, LUR renormings, renormings of Banach spaces, light groups, invariant renormings
Mathematical Subject Classification 2010
Primary: 22F50, 46B03
Milestones
Received: 21 December 2017
Revised: 25 May 2018
Accepted: 26 September 2018
Published: 16 September 2019
Authors
Leandro Antunes
Departamento de Matemática
Universidade Tecnológica Federal do Paraná
Campus Toledo
Toledo, PR
Brazil
Valentin Ferenczi
Departamento de Matematicá
Universidade de São Paulo
Instituto de Matemática e Estatística
São Paulo, SP
Brazil
Sophie Grivaux
CNRS, Univ. Lille, UMR 8524 - Laboratoire Paul Painlevé
France
Christian Rosendal
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
Chicago, IL
United States